Kavli Affiliate: Ran Wang
| First 5 Authors: Bin Qian, Min Wang, Ran Wang, Yimin Xiao,
| Summary:
Consider the nonlinear stochastic heat equation $$
frac{partial u (t,x)}{partial t}=frac{partial^2 u (t,x)}{partial x^2}+
sigma(u (t,x))dot{W}(t,x),quad t> 0,,
xin mathbb{R}, $$ where $dot W$ is a Gaussian noise which is white in time
and has the covariance of a fractional Brownian motion with Hurst parameter
$Hin(frac 14,frac 12)$ in the space variable. When $sigma(0)=0$, the
well-posedness of the solution and its H"older continuity have been proved by
Hu et al. cite{HHLNT2017}. In this paper, we study the asymptotic properties
of the temporal gradient $u(t+varepsilon, x)-u(t, x)$ at any fixed $t ge 0$
and $xin mathbb R$, as $varepsilondownarrow 0$. As applications, we deduce
Khintchine’s law of iterated logarithm, Chung’s law of iterated logarithm, and
a result on the $q$-variations of the temporal process ${u(t, x)}_{t ge 0}$,
where $xin mathbb R$ is fixed.
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